﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.37,0:00:08.93,Default,,0000,0000,0000,,Euler's number and the exponential function based on that number are of vital importance in the theory of differential equations. Dialogue: 0,0:00:08.93,0:00:16.06,Default,,0000,0000,0000,,Whenever we are dealing with exponential growth or exponential decay, and whenever we deal with oscillations, Dialogue: 0,0:00:16.07,0:00:22.10,Default,,0000,0000,0000,,Euler's number plays a big role. This is a brief and informed introduction to that topic, for which we need a minimum Dialogue: 0,0:00:22.10,0:00:29.17,Default,,0000,0000,0000,,amount of prerequisites. In particular, you should know that five to the first power equals five. Dialogue: 0,0:00:29.17,0:00:38.63,Default,,0000,0000,0000,,Five to the zeroth power equals one. Five to the minus second power equals one over 25. Dialogue: 0,0:00:38.63,0:00:43.60,Default,,0000,0000,0000,,Five to the power of one half equals the square root of five. Dialogue: 0,0:00:43.60,0:00:53.57,Default,,0000,0000,0000,,Five to the third power times five to the fourth power equals five to the power of three plus four--seven. Dialogue: 0,0:00:53.57,0:01:01.57,Default,,0000,0000,0000,,And five to the third power to the fourth power equals five to the power of 12--three times four. Dialogue: 0,0:01:01.57,0:01:04.93,Default,,0000,0000,0000,,This is what you need to know about powers of numbers. Dialogue: 0,0:01:04.93,0:01:13.03,Default,,0000,0000,0000,,In addition, you should have an idea about derivatives in this D, DX notation--for instance, the derivate of X squared Dialogue: 0,0:01:13.03,0:01:20.33,Default,,0000,0000,0000,,with respect to X equals 2X. And you should have an idea of how this is connected with the slope of the tangent line.